This inequality follows from the Littlewood's subordination
theorem (see, e.g., [1]}), but we shall give a simple direct proof.
If we as usual identify the function \(f(z)\in H^p\)  with its boundary value \(f(e^{i\theta })\) , then
\(\Vert f\circ \varphi _w\Vert _{H^p}^p=\frac{1}{2\pi }\int _{0}^{2\pi }|f(\varphi _w(e^{i\theta }))|^pd\theta =\frac{1}{2\pi i}\int _{0}^{2\pi }|f(\varphi _w(e^{i\theta }))|^p\frac{de^{i\theta }}{e^{i\theta }}.\) 
